I'm quite confused to be honest.
Could you please check that my examples are correct or not?

1) There is a gallup-poll company and this company measures that the citizens wants to participate on the elections or not.

They want to ask 2000 persons, and they choose an epsilon = 0.03.
Can I use Hoeffding's inequality for this issue at all?
2*exp(-2*0.03*0.03*2000) = 0.0546. Does it mean that they have 5% chance that their poll will close to the reality with an error of 3%?

If it's true, I'm very confused. Because it's also true in the US and in Monaco. In Monaco they asked every 2nd person, but in the USA they asked only very few persent of the population!

2) If we stay with the example of the bins:
It doesn't matter how many balls are there in the bin. It gives us an upper bound of the ratio of the red and green balls regarding the sample.
The thing which confuse me that the bin can have 100 balls or a million, sampling 10 balls we have the same upper bound to estimate the real distribution of the balls! I guess I don't understand something because it seems impossible.

I'm quite confused to be honest.
Could you please check that my examples are correct or not?

1) There is a gallup-poll company and this company measures that the citizens wants to participate on the elections or not.

They want to ask 2000 persons, and they choose an epsilon = 0.03.
Can I use Hoeffding's inequality for this issue at all?
2*exp(-2*0.03*0.03*2000) = 0.0546. Does it mean that they have 5% chance that their poll will close to the reality with an error of 3%?

If it's true, I'm very confused. Because it's also true in the US and in Monaco. In Monaco they asked every 2nd person, but in the USA they asked only very few persent of the population!

2) If we stay with the example of the bins:
It doesn't matter how many balls are there in the bin. It gives us an upper bound of the ratio of the red and green balls regarding the sample.
The thing which confuse me that the bin can have 100 balls or a million, sampling 10 balls we have the same upper bound to estimate the real distribution of the balls! I guess I don't understand something because it seems impossible.

In all the cases you mention, Hoeffding can be applied (if the sampling is done randomly). Hoeffding with replacement (put the ball back) and Hoeffding without replacement (never ask the same person) are slightly different, but similar in nature.

What Hoeffding guarantees is the worst case without considering the total number. When the size of the bin is smaller, you can surely get a bound that is tighter than Hoeffding without replacement. For instance, if you have 1000 balls and draw 1000 examples without replacement, you have zero chance to be any different.

I just recently attained a clear understanding of it myself (at least, I hope I did. The setting for it is a biased coin, with p the probability of Heads, and (1-p) the probability of Tails. A Bernoulli trial of length N with this coin is just N tosses of the coins. The Bernoulli outcomes (i.e., specific Heads-Tails sequences) of length N have the binomial distribution.

For each Bernoulli outcome of length N, we can ask: what is the "relative frequency" of Heads? (relative frequency = (# of Heads / N) ). So, the relative frequency of Heads can be regarded as a *random variable* on the probability space of Bernoulli trials of length N.

The Hoeffding inequality serves to answer the following question: Given an epsilon > 0, what is an upper bound on the probability that the above random variable deviates from p by at least epsilon?

For the purposes of this class, p is the probability that a hypothesis returns +1 ("Heads". This probability is more often denoted mu.

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